Research Article

Year 2024,
Volume: 45 Issue: 2, 366 - 378, 30.06.2024
### Abstract

### Keywords

### References

This study demonstrates the derivation of a two-step block scheme simulation through a linear block approach. The scheme's fundamental properties were thoroughly analyzed and found to fulfill all necessary conditions. The research focused on examining specific classes of oscillatory differential equations and comparing them to established methods. The findings indicate that the newly proposed methods exhibit superior accuracy and faster convergence compared to the existing methods investigated in this research. Consequently, the results highlight the improved precision and quicker convergence achieved with the new method. All computations were executed using Maple 18 software

Simulation Linear block approach Properties of the schemes Oscillatory differential equation Accuracy and convergence

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- [2] Raymond, D., Kyagya, T. Y., Sabo, J., Lydia, A., Numerical application of higher order linear block scheme on testing some modeled problems of fourth order problem, African Scientific Reports, 2(1) (2023).
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- [4] Sunday, J., James, A. A., Odekunle, M. R., Adesanya, A. O., Chebyshevian basis function-type block method for the solution of first-order initial value problems with oscillating solutions, J. Math. Comput. Sci., 5 (2015) 462- 472.
- [5] Sunday, J., Odekunle, M. R., Adesanya A. O., James, A. A., Extended block integrator for first-order stiff and oscillatory differential equations, Amer. J. Comput. Appl. Math. 3 (2013) 283-290.
- [6] James, A. A., Adesanya, A. O., Sunday, J., Yakubu, D. G., Half-step continuous block method for the solution of modeled problems of ordinary differential equations, American Journal of Computational Mathematics, 3 (2013) 261- 268.
- [7] Skwame, Y., Sabo, J., Kyagya, T. Y., The constructions of implicit one-step block hybrid methods with multiple off-grid points for the solution of stiff differential equations, Journal of Scientific Research and Report, 16(1)(2017) 1-7.
- [8] Ayinde, A. M., A. A. Ishaq, A. A., T. Latunde, T., Sabo, J., Efficient numerical approximation methods for solving high-order integro-differential equations, Caliphate Journal of Science & Technology, 2(3)(2021) 188-195.
- [9] Omar, Z., Adeyeye, A., Numerical solution of first order initial value problems using a self-starting implicit two-step Obrechkoff-type block method, Journal of Mathematic and Statistics, 12(2)(2016) 127-134.
- [10] Shokri, A., Shokri, A. A., The new class of implicit L-stable hybrid Obrechkoff method for the numerical solution of first order initial value problems, Comput. Phys. Commun. 184(2013) 529-530.
- [11] Ayinde, A.M, James, A.A, Ishaq, A.A., Oyedepo, T. , A new numerical approach using Chebyshev third kind polynomial for solving integro-differential equations of higher order, Gazi University Journal of Science, Part A: Engineering and Inovation, 9(3)(2022) 259 -266.
- [12] Oyedepo, T., Taiwo, O.A., Adewale, A.J., Ishaq, A.A., Ayinde, A.M., Numerical solution of system of linear fractional integro-differential equations by least squares collocation Chebyshev technique, Mathematics and Computational Sciences, 3(2) (2022) 10-21.
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- [14] Oyedepo, T., Ayinde, A.M., E.N. Didigwu, E.N., Vieta-Lucas polynomial computational technique for Volterra integro-differential equations, Electronic Journal of Mathematical Analysis and Applications, 12(1) (2024) 1-8. Dio.10.21608/EJMAA.2023.232998.1064
- [15] Oyedepo, T., Ishola, C.Y., Ishaq, A.A., Ahmed, O.L., Ayinde, A.M., Computational algorithm for fractional Fredholm integro-differential equations, Kathmandu University Journal of Science, Engineering and Technology, 17(1) (2023), 1-7.
- [16] Oyedepo, T., Ayoade, A. A., Otaide, I.K., Ayinde, A. M., Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations, Journal of Natural Sciences and Mathematical Research, 8 (2)(2022) 103-110
- [17] Kida, M., Adamu, S., Aduroja, O. O., Pantuvo, T. P., Numerical solution of stiff and oscillatory problems using third derivative trigonometrically fitted block method, Journal of Nigerian Society of Physical Sciences, 4(1)(2022) 34-48.
- [18] Ajileye, G., James, A. A., Ayinde, A. M., Oyedepo, T., Collocation approach for the computational solution of Fredholm-Volterra fractional order of integro-differential equations, Journal of Nigerian Society of Physical Sciences, 4(2022) 1-7.
- [19] Hasni, M. M., Majid, Z. A., Senu, N., Numerical solution of linear dirichlet two-point boundary value problems using block method, Int. J. Pure Applied Math., 85(3)(2013) 495-506.
- [20] Badmus, A. M., Yahaya, Y. A., Pam, Y. C., Adams type hybrid block methods associated with Chebyshev polynomial for the solution of ordinary differential equations, British J. Math. Comput. Sci. 6(6)(2015), 465-472.
- [21] Ali, K.K., Mehanna, M.S. and Akbar, M.A., 2022. Approach to a (2+ 1)-dimensional time-dependent date-JimboKashiwara-Miwa equation in real physical phenomena, Appl. Comput. Math., 21(2)(2022) 193-206.
- [22] Akram, G., Elahi, Z. and Siddiqi, S.S., Use of Laguerre polynomials for solving system of linear differential equations, Appl. Comput. Math., 21(2)(2022) 137-146.
- [23] Iskandarov, S. and Komartsova, E., On the influence of integral perturbations on the boundedness of solutions of a fourth-order linear differential equation, TWMS J. Pure Appl. Math., 13(1)(2022) 3-9.
- [24] Ozdemir ME. New refinements of hadamard integral inequality via k-Fractional integrals for p- Convex Function. Turkish Journal of Science, 6(1)(2021) 1-5.
- [25] Qi, F. Necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic, Appl. Comput. Math., 21(1)(2022) 61-70.
- [26] Shaalini, V. J., Fadugba, S. E., A new multistep method for solving delay differential equations using Lagrange interpolation, Journal of Nigerian Society of Physical Sciences, 3 (2021) 159-164.
- [27] Obarhua, F. O., Adegboro, O.J. Order four continuous numerical method for solving general second order ordinary differential equations, Journal of Nigerian Society of Physical Sciences 3 (2021) 42.
- [28] Sabo, J., Skwame, Y., Kyagya, T. Y., Kwanamu, J. A. ,The direct simulation of third order linear problems on single step block method. Asian Journal of Research in Computer Science. 12(2)(2021) 1-12.
- [29] Dahlquist, G. G., Numerical integration of ordinary differential equations, Mathematica Candinavica, 4(1956) 33-53.
- [30] Sunday, J., On Adomian decomposition method for numerical solution of ODEs arising from the natural laws of growth and decay, The Pacific Journal of Science and Technology, 12(1)(2011) 237-243.
- [31] Okunuga, S. A., Sofoluwe, A. B., Ehigie, J. O., Some block numerical schemes for solving initial value problems in ODEs, Journal of Mathematical Sciences, 2(1)(2013) 387-402.
- [32] Yan, Y.L., Numerical methods for differential equations, City University of Hong Kong Kowloon (2011).

Year 2024,
Volume: 45 Issue: 2, 366 - 378, 30.06.2024
### Abstract

### References

- [1] Ajileye, G., Aminu, F. A., A Numerical method using collocation approach for the solution of Volterra-Fredholm integro-differential equations, African Scientific Reports, 1(3) (2022) 206.
- [2] Raymond, D., Kyagya, T. Y., Sabo, J., Lydia, A., Numerical application of higher order linear block scheme on testing some modeled problems of fourth order problem, African Scientific Reports, 2(1) (2023).
- [3] Henrici, P., Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., Hoboken, NJ, USA, (1969).
- [4] Sunday, J., James, A. A., Odekunle, M. R., Adesanya, A. O., Chebyshevian basis function-type block method for the solution of first-order initial value problems with oscillating solutions, J. Math. Comput. Sci., 5 (2015) 462- 472.
- [5] Sunday, J., Odekunle, M. R., Adesanya A. O., James, A. A., Extended block integrator for first-order stiff and oscillatory differential equations, Amer. J. Comput. Appl. Math. 3 (2013) 283-290.
- [6] James, A. A., Adesanya, A. O., Sunday, J., Yakubu, D. G., Half-step continuous block method for the solution of modeled problems of ordinary differential equations, American Journal of Computational Mathematics, 3 (2013) 261- 268.
- [7] Skwame, Y., Sabo, J., Kyagya, T. Y., The constructions of implicit one-step block hybrid methods with multiple off-grid points for the solution of stiff differential equations, Journal of Scientific Research and Report, 16(1)(2017) 1-7.
- [8] Ayinde, A. M., A. A. Ishaq, A. A., T. Latunde, T., Sabo, J., Efficient numerical approximation methods for solving high-order integro-differential equations, Caliphate Journal of Science & Technology, 2(3)(2021) 188-195.
- [9] Omar, Z., Adeyeye, A., Numerical solution of first order initial value problems using a self-starting implicit two-step Obrechkoff-type block method, Journal of Mathematic and Statistics, 12(2)(2016) 127-134.
- [10] Shokri, A., Shokri, A. A., The new class of implicit L-stable hybrid Obrechkoff method for the numerical solution of first order initial value problems, Comput. Phys. Commun. 184(2013) 529-530.
- [11] Ayinde, A.M, James, A.A, Ishaq, A.A., Oyedepo, T. , A new numerical approach using Chebyshev third kind polynomial for solving integro-differential equations of higher order, Gazi University Journal of Science, Part A: Engineering and Inovation, 9(3)(2022) 259 -266.
- [12] Oyedepo, T., Taiwo, O.A., Adewale, A.J., Ishaq, A.A., Ayinde, A.M., Numerical solution of system of linear fractional integro-differential equations by least squares collocation Chebyshev technique, Mathematics and Computational Sciences, 3(2) (2022) 10-21.
- [13] Oyedepo, T., Ayoade, A.A., Ajileye, G., Ikechukwu, N.J., Legendre computational algorithm for linear integro-differential equations, Cumhuriyet Science Journal, 44(3) (2023) 561-566.
- [14] Oyedepo, T., Ayinde, A.M., E.N. Didigwu, E.N., Vieta-Lucas polynomial computational technique for Volterra integro-differential equations, Electronic Journal of Mathematical Analysis and Applications, 12(1) (2024) 1-8. Dio.10.21608/EJMAA.2023.232998.1064
- [15] Oyedepo, T., Ishola, C.Y., Ishaq, A.A., Ahmed, O.L., Ayinde, A.M., Computational algorithm for fractional Fredholm integro-differential equations, Kathmandu University Journal of Science, Engineering and Technology, 17(1) (2023), 1-7.
- [16] Oyedepo, T., Ayoade, A. A., Otaide, I.K., Ayinde, A. M., Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations, Journal of Natural Sciences and Mathematical Research, 8 (2)(2022) 103-110
- [17] Kida, M., Adamu, S., Aduroja, O. O., Pantuvo, T. P., Numerical solution of stiff and oscillatory problems using third derivative trigonometrically fitted block method, Journal of Nigerian Society of Physical Sciences, 4(1)(2022) 34-48.
- [18] Ajileye, G., James, A. A., Ayinde, A. M., Oyedepo, T., Collocation approach for the computational solution of Fredholm-Volterra fractional order of integro-differential equations, Journal of Nigerian Society of Physical Sciences, 4(2022) 1-7.
- [19] Hasni, M. M., Majid, Z. A., Senu, N., Numerical solution of linear dirichlet two-point boundary value problems using block method, Int. J. Pure Applied Math., 85(3)(2013) 495-506.
- [20] Badmus, A. M., Yahaya, Y. A., Pam, Y. C., Adams type hybrid block methods associated with Chebyshev polynomial for the solution of ordinary differential equations, British J. Math. Comput. Sci. 6(6)(2015), 465-472.
- [21] Ali, K.K., Mehanna, M.S. and Akbar, M.A., 2022. Approach to a (2+ 1)-dimensional time-dependent date-JimboKashiwara-Miwa equation in real physical phenomena, Appl. Comput. Math., 21(2)(2022) 193-206.
- [22] Akram, G., Elahi, Z. and Siddiqi, S.S., Use of Laguerre polynomials for solving system of linear differential equations, Appl. Comput. Math., 21(2)(2022) 137-146.
- [23] Iskandarov, S. and Komartsova, E., On the influence of integral perturbations on the boundedness of solutions of a fourth-order linear differential equation, TWMS J. Pure Appl. Math., 13(1)(2022) 3-9.
- [24] Ozdemir ME. New refinements of hadamard integral inequality via k-Fractional integrals for p- Convex Function. Turkish Journal of Science, 6(1)(2021) 1-5.
- [25] Qi, F. Necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic, Appl. Comput. Math., 21(1)(2022) 61-70.
- [26] Shaalini, V. J., Fadugba, S. E., A new multistep method for solving delay differential equations using Lagrange interpolation, Journal of Nigerian Society of Physical Sciences, 3 (2021) 159-164.
- [27] Obarhua, F. O., Adegboro, O.J. Order four continuous numerical method for solving general second order ordinary differential equations, Journal of Nigerian Society of Physical Sciences 3 (2021) 42.
- [28] Sabo, J., Skwame, Y., Kyagya, T. Y., Kwanamu, J. A. ,The direct simulation of third order linear problems on single step block method. Asian Journal of Research in Computer Science. 12(2)(2021) 1-12.
- [29] Dahlquist, G. G., Numerical integration of ordinary differential equations, Mathematica Candinavica, 4(1956) 33-53.
- [30] Sunday, J., On Adomian decomposition method for numerical solution of ODEs arising from the natural laws of growth and decay, The Pacific Journal of Science and Technology, 12(1)(2011) 237-243.
- [31] Okunuga, S. A., Sofoluwe, A. B., Ehigie, J. O., Some block numerical schemes for solving initial value problems in ODEs, Journal of Mathematical Sciences, 2(1)(2013) 387-402.
- [32] Yan, Y.L., Numerical methods for differential equations, City University of Hong Kong Kowloon (2011).

There are 32 citations in total.

Primary Language | English |
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Subjects | Applied Mathematics (Other) |

Journal Section | Natural Sciences |

Authors | |

Publication Date | June 30, 2024 |

Submission Date | August 18, 2023 |

Acceptance Date | January 3, 2024 |

Published in Issue | Year 2024Volume: 45 Issue: 2 |